A Level Maths 9709 · Paper 6 Probability & Statistics 2 · 6.1 · ~4 min read
The Poisson distribution
Modelling random events with Po(λ), its conditions, scaling λ to a new interval, mean equal to variance, and using the Poisson to approximate the binomial and the normal to approximate the Poisson.
Po(λ)ConditionsP(X = r)Scaling λMean = variancePoisson approximation to binomialNormal approximation to PoissonContinuity correction
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Paper tag = where it is examinedFORMULA LIST (MF19) given in the examM1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end
In 30 seconds
- Po(λ) counts random events in a fixed interval of time or space: events happen singly, independently, and at a constant average rate λ.
- Mean = variance = λ. If a sample has mean close to its variance, a Poisson model is plausible.
- Change of interval: scale λ in proportion. 4 per hour means 2 per half hour.
- Approximations: B(n, p) ≈ Po(np) when n is large (n > 50) and np < 5; Po(λ) ≈ N(λ, λ) when λ > 15, with a continuity correction.
By the end you can
- Use the Poisson distribution as a model and state its conditions in context.
- Calculate Poisson probabilities, including for a scaled interval.
- Use E(X) = Var(X) = λ.
- Use the Poisson approximation to the binomial and the normal approximation to the Poisson.
1The Poisson model
Po(λ)
P(X = r) = e−λ λʳ/r! , r = 0, 1, 2, … ; E(X) = Var(X) = λ FORMULA LIST (MF19)
λ is the mean number of events in the interval- Conditions in context: "the calls occur randomly, independently, one at a time, at a constant average rate". Give them in the context of the question.
Worked example
X ~ Po(3.2). Find (a) P(X = 2), (b) P(X ≥ 3).
- (a) e−3.2 × 3.2²/2 = 0.209 M1 A1
- (b) 1 − [P(0) + P(1) + P(2)] = 1 − e−3.2(1 + 3.2 + 5.12) M1
- = 0.620 A1
Worked example
Calls arrive at an average rate of 4 per hour. Find the probability of at most 1 call in a 30-minute period.
- λ = 2 for 30 minutes B1
- e−2(1 + 2) = 3e−2 M1
- = 0.406 A1
Common mistakeNot scaling λ, or rescaling the probability instead of λ. The 2024 Paper 6 report also asked for conditions "in context": a bare "independent" scores nothing.
2Approximations
Poisson for binomial; normal for Poisson
- Poisson for binomial: many trials with a small success probability. Use λ = np.
- Normal for Poisson: λ > 15, Y ~ N(λ, λ), and continuity correction because X is discrete.
Worked example
X ~ B(200, 0.015). Use a suitable approximation to find P(X ≤ 2).
- n > 50, np = 3 < 5, so use Po(3) B1
- e−3(1 + 3 + 4.5) M1
- = 0.423 A1 (exact binomial: 0.421)
Worked example
The number of emails a day is Po(25). Use a suitable approximation to find P(X > 30).
- λ > 15: Y ~ N(25, 25) B1
- P(Y > 30.5): z = 5.5/5 = 1.1 M1
- 1 − 0.8643 = 0.136 A1
Common mistakeUsing σ = 25 instead of √25 = 5, or forgetting the continuity correction.
Linking questionThe binomial is in Discrete random variables; sums of Poisson variables are in Linear combinations; and Poisson hypothesis tests are in Hypothesis tests.
Test yourself — cover the answers first
Quick check
- X ~ Po(5). Find Var(X).
- X ~ Po(2). Find P(X = 0).
- A machine averages 6 faults per week. What is λ for 2 weeks?
- When can B(n, p) be approximated by a Poisson?
- Continuity-correct P(X ≥ 20) for a normal approximation.
Exam-style
- X ~ Po(1.5). Find P(X ≤ 2 ǀ X ≥ 1).
- Typing errors occur at 0.8 per page. Find the probability of more than 3 errors in 5 pages.
- Sample data: mean 3.1, variance 7.2. Is a Poisson model suitable?
Show answers
- 5
- e−2 = 0.135
- 12
- n large (> 50) and np < 5
- P(Y > 19.5)
- [P(1) + P(2)]/[1 − P(0)] = (0.3347 + 0.2510)/0.7769 = 0.754
- λ = 4: 1 − e−4(1 + 4 + 8 + 10.667) = 0.567
- no, the variance is much larger than the mean
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